What is the derivative of ln(x+1/(ln(x))) ?

The derivative of ln(x+1/(ln(x))) is 1/x-1/(xln^2(x))

What is the first derivative of ln(x+1/(ln(x))) ?

The first derivative of ln(x+1/(ln(x))) is 1/x-1/(xln^2(x))

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\frac{d}{d x} (ln (x) + \frac{1}{ln ( x )})

\frac{1}{x} - \frac{1}{x ln ^{2} ( x )}

Solution steps

\frac{d}{d x} (ln (x) + \frac{1}{ln ( x )})

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d x} (ln (x)) + \frac{d}{d x} (\frac{1}{ln ( x )})

\frac{d}{d x} (ln (x)) = \frac{1}{x}

\frac{d}{d x} (\frac{1}{ln ( x )}) = - \frac{1}{x ln ^{2} ( x )}

= \frac{1}{x} - \frac{1}{x ln ^{2} ( x )}

What is the derivative of ln(x+1/(ln(x))) ?
The derivative of ln(x+1/(ln(x))) is 1/x-1/(xln^2(x))
What is the first derivative of ln(x+1/(ln(x))) ?
The first derivative of ln(x+1/(ln(x))) is 1/x-1/(xln^2(x))